monolayersinopticallattices arxiv:1508.06201v2 [cond-mat.mes … · 2016. 9. 1. · e-mail:...

arX

iv:1

508.

0620

1v2

[co

nd-m

at.m

es-h

all]

7 O

ct 2

015 Subharmonic Shapiro steps of sliding colloidal

monolayers in optical lattices

Stella V. Paronuzzi Ticco1,2, Gabriele Fornasier1, Nicola

Manini1,2, Giuseppe E. Santoro2,3,4, Erio Tosatti2,3,4, and Andrea

Vanossi2,3

1 Dipartimento di Fisica and CNR-INFM, Universita di Milano, Via Celoria 16,

20133 Milano, Italy2 International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136

Trieste, Italy3 CNR-IOM Democritos National Simulation Center, Via Bonomea 265, 34136

Trieste, Italy4 International Center for Theoretical Physics (ICTP), P.O.Box 586, I-34014 Trieste,

Italy

E-mail: [email protected]

Abstract. We investigate theoretically the possibility to observe dynamical mode

locking, in the form of Shapiro steps, when a time-periodic potential or force

modulation is applied to a two-dimensional (2D) lattice of colloidal particles that are

dragged by an external force over an optically generated periodic potential. Here

we present realistic molecular dynamics simulations of a 2D experimental setup,

where the colloid sliding is realized through the motion of soliton lines between

locally commensurate patches or domains, and where the Shapiro steps are predicted

and analyzed. Interestingly, the jump between one step and the next is seen to

correspond to a fixed number of colloids jumping from one patch to the next, across

the soliton line boundary, during each AC cycle. In addition to ordinary “integer”

steps, coinciding here with the synchronous rigid advancement of the whole colloid

monolayer, our main prediction is the existence of additional smaller “subharmonic”

steps due to localized solitonic regions of incommensurate layers executing synchronized

slips, while the majority of the colloids remains pinned to a potential minimum.

The current availability and wide parameter tunability of colloid monolayers makes

these predictions potentially easy to access in an experimentally rich 2D geometrical

configuration.

PACS numbers: 82.70.Dd,83.85.Vb,68.35.Af

http://arxiv.org/abs/1508.06201v2

Subharmonic Shapiro steps of sliding colloidal monolayers in optical lattices 2

1. Introduction

Driven nonlinear systems can show widely intriguing patterns of varied and often

unexpected dynamical behavior. An especially interesting case is that of many identical

interacting and crystallized particles moving in a periodic external potential: a common

situation, e.g. for adatom structures sliding on a crystalline surface. In the special

case when, besides the main time-independent force which causes the sliding, an

additional external ”AC” force component is present and oscillates periodically in time,

synchronization phenomena are known to occur, depending on the relative importance

of the interactions and on the geometrical arrangement of the particles.

Synchronization phenomena of this kind have long been described using simple, low-

dimensional, phenomenological models, which often suffice to capture the main features

of the complex dynamics involved. The basic example is provided by the simple yet

fundamental one-dimensional (1D) Frenkel-Kontorova (FK) chain model [1, 2, 3] whose

soliton, or kink pattern is known to exhibit a nontrivial intermittent dynamics, when

additional time-periodic forces act on the otherwise steadily sliding chain [4]. This simple

1D model however is not generally sufficient to describe in full the complex dynamics

of atoms at the interface of two materials in relative motion. Experimental studies

of two-dimensional (2D) arrays of mutually repelling colloids pushed across a periodic

“corrugation” potential landscape have recently been carried out, providing a closer

analog of the sliding of crystalline interfaces [5, 6, 7, 8]. Before sliding, the observed 2D

patterns of misfit dislocation lines (referred to as solitons or kinks) are very much akin

to the Moire patterns formed by atomic overlayers over crystal surfaces. Besides their

surface physics analog, the colloid systems have an interest on their own because they

provide a ready parameter-controlled system for the study of friction between crystalline

surfaces [9, 7, 10].

Here we explore, ahead of experiments, just very recently available in a simpler 1D

geometry [11], a situation where synchronization, whence the existence and the actual

dynamical nature of Shapiro steps, is sought by oscillating in time either the pushing

force or the amplitude of the (x, y) periodic corrugation potential. In particular we focus

on the second, corresponding to an ideal setup where colloids thread a partially time-

modulated spatially-periodic optical potential, and investigate in particular the effects

of the mismatch between the colloid-colloid average spacing of the 2D colloid crystal

and the periodicity of the optical potential.

Shapiro steps were initially observed [12] in the context of Josephson junctions.

The crucial ingredient at the start was Josephson’s prediction [13] of a coherent

current Ic sinφ flowing between two superconductors separated by a thin insulating

barrier in presence of a phase difference φ = φ1 − φ2 between the two superconductor

order parameters. Moreover, a potential difference V = h2e

dφdt

was predicted to be

associated to a phase φ that changes in time, with the remarkable consequence, known

as AC Josephson effect, that the coherent current would oscillate with a frequency

ω0 = 2eVdc/h in presence of a constant DC bias Vdc. The fingerprint of ω0 can be


detected through an ingenious resonance experiment [12], in which a current I(t) =

Idc + Iac sin(ωt) is passed through the junction. A simple model [14, 15] capturing

resistive and capacitive effects in the junction would allow us to write:

hC

2e

d2φ

dt2+

h

2eR

dφ

dt+ Ic sinφ = I(t) , (1)

where C is the capacitance of the junction, and hC2e

d2φdt2

= C dVdt

the corresponding current,

while h2eR

dφdt

= VR

accounts for the resistive part of the current, due to quasi-particles.

In general, a φ(t) = ω0t + ∆φ(t) where ∆φ(t) is a bounded function, implies a (time-

averaged) DC voltage Vdc =hω0

2eacross the junction. In a non-resonant situation, sinφ(t)

oscillates in an aperiodic way with zero average, 〈sinφ(t)〉time = 0; hence the (time-

averaged) DC voltage predicted by Eq. (1) follows the usual Ohmic I-V characteristics,

Vdc = RIdc. Resonances instead can make 〈sinφ(t)〉time 6= 0 by the following phase-

locking mechanism [16]: the bounded variation ∆φ(t) becomes a periodic function of

time, characterized by the frequency ω of the external driving, while the slope of

φ(t) — the frequency ω0 — exactly matches a multiple of the driving frequency ω:

ω0 = nω. In such resonant situations, sinφ(t) becomes periodic and with non-zero

average, 〈sinφ(t)〉time 6= 0, implying that the constant terms in Eq. (1) obtained by

averaging over a period sum up to:

Vdc

R+ Ic〈sinφ(t)〉time = Idc . (2)

In essence, at resonances, Idc can be changed by an amount ∼ Ic〈sinφ(t)〉time without

affecting the voltage bias Vdc: these are the Shapiro steps in the I-V characteristics,

where the the voltage step is strictly quantized through a finite range of current.

A mechanical analogy of the Josephson junction model of Eq. (1) is straightforward,

with the phase difference φ(t) playing the role of a coordinate x(t), the voltage V ∝ dφ/dt

that of the velocity of a particle moving in a washboard potential, and the current

I(t) the role of an external force. Shapiro steps have then a natural translation into

“quantized” steps of the velocity of the moving particle for a finite range of applied force,

when the washboard frequency goes in resonance with the AC perturbing frequency ω.

This resonance phenomenon is robust, and one could for instance periodically modulate

the amplitude of the space-periodic corrugation potential [the analog of Ic in Eq. (1)]

rather than the external force, and the same physics would follow.

As is shown in Eq. (1), the Shapiro physics involves a single degree of

freedom. However, in systems of many interacting particles, crystallized but not rigid,

synchronization phenomena of the Shapiro kind are likely to give rise to additional

nontrivial dynamical effects due to concerted multi-particle motion. The strong interest

in synchronization phenomena in multiple physical contexts has generated numerous

reseach works in recent years, among which we recall Refs. [11, 17, 18, 19, 20, 21, 22,

23, 24, 25]. The system where we propose to explore the Shapiro physics consists of a

collection of repulsive colloidal particles dragged by an external force over a spatially

periodic corrugation potential, whose amplitude includes both a static and a time-

oscillating part. When the frequency ω of the oscillation is a multiple of the characteristic


(washboard) frequency ω0, the forced sliding motion of the 2D colloid lattice can give rise

to Shapiro steps. Combined experimental and simulation studies already addressed in

the past the effect of synchronization and the ensuing Shapiro steps in Brownian particle

dynamics [24]. Other simulation work [26] addressed the case where the 2D sliding

lattice and the periodic potential are fully matched, in addition to ratcheting conditions

in mixtures. Very recently, the microscopic dynamics underlying mode locking in a

colloidal model system has been recorded in a simple, yet noteworthy, 1D experimental

setup [11]. In this work we study the feasibility of observing Shapiro steps under

reasonably realistic conditions in 2D sliding colloid monolayers, under the wider range

of conditions that can be realized experimentally. More specifically, we will describe a

planar system of monodisperse repulsive particles, forming a 2D triangular lattice with

spacing acoll, sliding within a periodic corrugation potential due to an optical lattice, also

triangular with spacing alas, whose amplitude is partly modulated periodically in time,

under various conditions as determined by different choices of the commensurability

ratio ρ = alas/acoll. Using classical molecular dynamic we will monitor the individual

motion of the particles as well as that of their center of mass (CM) under the action of

a DC force. As a function of this force the forward CM velocity of the colloidal particles

should, owing to Shapiro synchronization, become ’quantized’ in step-like structures as

a function of the applied force. Our aim is therefore to seek, detect and characterize

these step structures in relation to the absence (in the fully matched, commensurate

case ρ = 1) or presence, e.g., for ρ < 1 of pre-existing soliton-like defects or kinks due

to mismatched relative spacings of the colloidal lattice and of the periodic potential.

2. The model

Molecular dynamics (MD) simulations are based on the same model already introduced

[10] to describe the motion of mutually repulsive charged colloidal particles driven across

a periodic potential generated by a light interference pattern, as realized in experiments

by Bohlein et al. [7, 8]. In short, we describe the charged colloidal particles as classical

point-like objects, whose dynamics is affected by their mutual repulsion, the action of

external forces, plus the interaction with the viscous fluid where they are immersed.

The equation of motion for the j-th particle is

mrj(t) + η(rj(t)− vdx) = −∇rj(U2 + Uext) + fj(t) , (3)

where rj is the position of the j-th colloid in the 2D plane where it can move, η is

the friction coefficient determined by the effective viscosity of the fluid in which the

colloids are immersed, and vd is the fluid drift velocity, giving rise to a Stokes driving

force F = ηvd, pushing forward all colloidal particles. U2 is the two-body inter-particle

potential; Uext is the corrugation potential describing the interaction with the spatially

periodic light-field pattern. In this work we especially focus on the effect of time-

periodic oscillations of the amplitude of Uext. The finite-temperature Brownian motion

of colloids is introduced in a standard Langevin approach, involving the viscous friction


ρ η N Nminima Q λD Lx Ly

1 28 392 392 1013 0.03 14 14√3

14/15 28 392 450 1013 0.03 14 14√3

3/4 28 450 800 1013 0.03 15 15√3

Table 1: Adopted simulation parameters, expressed in model units [10]. For example,

the simulation-cell sides Lx and Ly are given in units of the average colloid lattice spacing

acoll. Nminima is the number of identical local minima of the corrugation function W (r)

contained in a simulation cell.

term η(rj(t) − vdx), plus the Gaussian random force fj(t) [27]. Due to the low overall

velocity (vd ≃ 10 µms−1) of the colloidal particles, the inertial term can be neglected,

and an overdamped diffusive motion is reasonably assumed, with a sufficiently large η.

In our work η = 28 expressed in the same model units described in Ref. [10]. Typical

simulation parameters are reported in Table 1.

The two-body interaction potential is

U2 =N∑

j<j′VYuk(|rj − rj′|) , (4)

where the screened Coulomb repulsion V is a Yukawa potential:

VYuk(r) =Q

rexp(−r/λD) . (5)

The average nearest-neighbor separation of colloids in Ref. [7] is r = acoll ≃ 5.7 µm ≃30λD. Under confinement, this repulsion establishes a triangular lattice of colloids,

which can be thought as reasonably defect free at temperatures not too high and sizes

sufficiently small to render the possibility of Nelson-Halperin dislocations [28] irrelevant

to the present case.

The one-body term

Uext =N∑

j

Vext(rj, t) , (6)

describes the interaction of the colloids with the 2D spatially periodic potential,

representing the optical lattice corrugation. Its actual form, determined by Vext(r),

could in principle be shaped with some freedom, but the simplest sinusoidal form is

sufficient to describe the main phenomena. We thus take:

Vext(r, t) = [V0 +∆0 sin(ωt)]W (r) , (7)

where

W (r) = −1

9

[

3 + 4 cos

(

2πry√3 alas

)

cos(

2πrxalas

)

+ 2 cos

(

4πry√3 alas

)]

, (8)

a unit-amplitude eggcarton-type periodic potential of 6-fold symmetry representing the

optical-lattice triangular corrugation. The time-dependent amplitude V0 + ∆0 sin(ωt)


gives to the corrugation an additional sinusoidal time dependence that can induce

synchronization effects. Its amplitude 2∆0 yields a total corrugation ranging between

V0 − ∆0 and V0 + ∆0 with an AC modulation frequency ν = ω/(2π). The minimum

force required to dislodge an isolated colloid at T = 0 along direction x is at time t

Fs1 =8π

9

V0 +∆0 sin(ωt)

alas. (9)

The quantity 8V0/(9alas) is used as our unit of force. In general, the spacing alas of the

corrugation lattice potential W (x, y) can be tuned in such a way to be either matched

or mismatched with the colloidal lattice. In mismatched configurations, one observes,

both in simulations [10] and in experiments [7], the formation of misfit dislocations also

called topological solitons, or kinks/antikinks in the language of the FK model [29]. Their

existence and their motion under the external force dominates the frictional properties

of the sliding lattice, since it provides the mechanism for mass transport [30].

As for the oscillation frequency, it needs to be low enough for the overdamped

colloidal system to be allowed enough time to follow adiabatically the AC modulation

in the corrugation-amplitude oscillation. The specific condition is ν ≪ η/m, and we use

ν < 0.1 in model units. Shapiro-step structures can arise out of synchronization of the

washboard frequency ω0 of the colloids driven by F over the corrugation potential, with

the time-modulation frequency ω of Eq. (7). At finite temperature (e.g. in experimental

conditions), ν must also not to be so small that particle random diffusion overshadows

the ordered synchronized crossing of potential energy barrier.

Our protocol begins with a preliminary relaxation preparing the sample in its

equilibrium, force-free initial state. Sliding simulations are then conducted by applying

a force F = ηvd along x on each particle, similar to the viscous drag of the flowing fluid

in experiments. The static force F is slowly ramped upwards in small steps ∆F , so as

to explore the desired force range. At each force value, after an initial transient lasting

between 1 and 10 oscillation periods, the particle CM displacement ∆xcm is extracted

for the successive simulation time τ , typically a large integer (of order 100) number

of periods ν−1 of the time-dependent perturbation. We then record the average speed

vcm = ∆xcm/τ for each value of the driving force F .

Experimental conditions [7] actually involve an inhomogeneous 2D configuration,

whereby only a circular central portion of the colloid monolayer raft is submitted to

the laser field with its associated optical lattice potential. However, only phenomena in

the central part of this circle are eventually studied, and there the colloids are virtually

homogeneous. Therefore, we conduct the present study in a fully homogeneous 2D

colloid lattice with periodic boundary conditions (PBC) qualitatively represented in

Fig. 1. Given a cell size and a particle number, we will simulate either the commensurate

and fully matched geometry (ρ = 1), or mismatched underdense geometries (ρ < 1), by

tuning appropriately the corrugation lattice spacing alas relative to that of the colloid

lattice, acoll.


Figure 1: The Lx × Ly = 15 acoll × 15√3 acoll rectangular simulation cell (solid line)

adopted for the ρ = 3/4 simulations. In this initial snapshot, the 450 particles still form

a perfect triangular lattice.

3. Perfect lattice matching

We begin with the fully matched commensurate geometry where, at ρ = 1, the two

triangular lattices coincide, and no preliminary relaxation is needed, each particle falling

in a potential minimum. We then turn on the sliding force with a time-dependent

modulation of amplitude ∆0 = 40%V0, with an intermediate corrugation strength

V0 = 0.5, at T = 0. The average CM velocity as a function of the external force

F is presented in Fig. 2. As a function of the sliding force, it shows Shapiro steps,

i.e. plateaus in the average CM velocity. These steps correspond to integer multiples

of νalas in the force-velocity response. In these T = 0 simulations the motion of the

colloids is deterministic, leading to perfectly flat steps with null error bars, as long as

the CM speed is averaged over an integer number of periods ν−1. The reference velocity

νalas is that of a particle advancing by one corrugation spacing alas in one oscillation

period ν−1. Depending on the applied DC force, all particles advance together by an

integer number of lattice spacings alas at each period of the AC modulation. As a

function of ν both the velocity jump between successive Shapiro steps, and the force

step width increase proportionally to ν, as expected. Correspondingly, the number of

steps in a given force range decreases for increasing ν. In the large force regime, for

F > 8π9(V0 + ∆0)/alas ≃ 1.95, the steps rapidly decrease in width, and merge into a


0

2

4

6

8

10

vcm

/ (

ν a

las)

ν=0.01

0

2

4

vcm

/ (

ν a

las)

ν=0.02

0 1 2 3

F (DC Force)

0

1

2

3

vcm

/ (

ν a

las)

ν=0.03

Figure 2: T = 0 Shapiro steps observed in the CM velocity parallel to the applied force

obtained for a fully commensurate colloid lattice (alas = acoll, ρ = 1) with a corrugation

amplitude V0 = 0.5, a modulation amplitude ∆0 = 0.2, for values of the frequency

ν = 0.01, 0.02, and 0.03 (in model units). The velocity is scaled by νalas, to emphasize

the proportionality of the exactly quantized steps to the modulation frequency.

smooth velocity increase proportional to force, characteristic of a viscous regime, similar

to the non-modulated ∆0 = 0 dynamics [7, 10].

In this zero-temperature lattice-matched system, all colloids execute the same

movement to negotiate the crossing of the same energy barriers. As a result, the

colloid-colloid mutual spacing remains constant, and the colloid-colloid internal force

does no work. The dynamics would be exactly the same if the colloidal particles were

not interacting, or if the simulation involved one particle only, moving in the same

potential-energy profile. As a check, we repeated the simulations with a single particle,

indeed recovering exactly the same patterns as in Fig. 2.

We explore now how the AC modulation amplitude ∆0 affects these Shapiro steps.

Figure 3 shows the velocity curves obtained with the same average corrugation amplitude

(V0 = 0.5) and AC modulation amplitude rising from the same initial value ∆0 = 40%V0

as in the central panel of Fig. 2 (ν = 0.02), to larger values ∆0 = 70%V0 and

∆0 = 100%V0, at the same frequency. For larger ∆0 the colloid system starts to move

earlier, i.e., at a smaller DC force than for a smaller ∆0. Note that at 100% relative

modulation amplitude, every oscillation period has a brief instant where the colloid

layer actually slides freely. Despite this free-sliding instant, the force needed for the

monolayer to depin and reach the first Shapiro step, even though smaller than at 40%

modulation amplitude, appears anyway to be nonzero. The reason for that is the finite

time it takes the colloid to drift across the momentarily turned off barrier to the next

substrate minimum, when the driving force F is small. This issue is discussed in detail

in Sect. 5 below.


0 0.5 1 1.5

F (DC Force)

0

1

2

vcm

/(ν

ala

s)

∆0 = 100% V

0

∆0 = 70% V

0

∆0 = 40% V

0

Figure 3: Shapiro steps for ν = 0.02, V0 = 0.5, for increasing AC modulation amplitude

∆0. Pluses: ∆0 = 40%V0, same as in the central panel of Fig. 2. Crosses: ∆0 = 70%V0.

Squares: ∆0 = 100%V0. The step heights are unchanged, their widths increase, and the

jump between successive steps becomes steeper for increasing ∆0.

0 0.5 1 1.5 2

F (DC Force)

0

0.5

1

1.5

2

vcm

/(ν

ala

s)

∆0 = 70% V

0

Figure 4: Finite-temperature Shapiro-step structure for a fully commensurate system.

Temperature T = Troom, and the other simulation conditions (ν = 0.02, V0 = 0.5,

∆0 = 70%V0) are the same as for the crosses data of Fig. 3. The effect of thermal

motion is barely detectable, apart from a slight rounding on the step edges. Thermal

fluctuations of the CM velocity, reflected by the error bars, are visible at rises between

steps, but negligible inside the steps.

3.1. Thermal effects

So far we worked at T = 0, because that regime yields a clearer picture. Next, however,

we can ask how the Shapiro steps might be affected by thermal fluctuations in the

experimental, room-temperature conditions. To address this question, we perform new

simulations at room temperature T = Troom (Troom = 0.04 in system units), still at

ρ = 1 and keeping all other parameters (mimicking the experimental colloid system of

Ref. [7]) the same. Figure 4 shows that the Shapiro-step structure is barely affected;

Troom is a very moderate temperature for this system. Thermal fluctuations (estimated

by the standard deviation of vcm obtained over several sliding simulations realized with


independent random-number sequences and different initial configurations ensuring a

better sampling of the thermal equilibrium state) are visible, as reflected by the error

bars, at jumps between steps. Thermal fluctuations produce a rounding of the plateau

onset, but are barely visible inside each plateau.

4. Mismatched lattice spacing

When the commensurability ratio ρ = alas/acoll differs from unity the colloid and

the corrugation lattices are mismatched. As we shall see, in this case the internal

structure of the sliding lattice plays a role, and the expected Shapiro-step phenomena

become more interesting. It must be clarified at the outset that the Shapiro steps

exist only if the system is pinned by a finite static friction force. For example, a

2D colloid lattice incommensurate with a weak periodic corrugation is an unpinned

system which can slide “superlubrically” without static friction, and will exhibit no

Shapiro steps. The same incommensurate lattice must however become pinned when

the corrugation amplitude exceeds a first-order depinning-pinning transition threshold

value [31], and here Shapiro steps can exist. This is the situation which we concentrate

upon here. We describe in detail two different examples of mismatched configurations,

ρ = 14/15 ≃ 0.93 and ρ = 3/4 = 0.75. Both are underdense systems compared to the

perfectly matched one with ρ = 1 considered in Sec. 3, and both are of course rationally

commensurate. However, the commensurabilty of ρ = 3/4 is so to speak stronger, that

in ρ = 14/15 weaker, the latter case in a sense closer to incommensurability, itself a

condition impossible to reach in a finite-size PBC realization. We will further assume

that the colloids and the corrugation lattice do not undergo a relative rotation, e.g.,

of the type studied in Refs. [32, 33, 34, 35, 36], and therefore remain aligned. With

that stipulation we adopt for ρ = 14/15 a rectangular periodic simulation cell involving

392 colloids threading 450 corrugation potential minima. With 14 colloidal particles

threading 15 potential wells along each principal direction, we obtain a widely spaced

hexagonal network of misfit dislocation lines (antisolitons), where vacancies segregate.

Details of this case will be shown later. For ρ = 3/4 we also adopt a rectangular

supercell, here involving 450 colloids and 800 potential minima, shown in Fig. 1. The

higher vacancy concentration leads to a greater mismatch and therefore a denser packing

of antisolitons.

In these mismatched systems we must first of all identify the corrugation amplitude

at which, as was mentioned above, a pinning/depinning transition takes place in the

absence of AC modulation (∆0 = 0). In the limit of infinitesimal applied force,

this transition was recently discovered and characterized by Mandelli et al. [31].

Ignoring here the weak 14/15 commensurability and taking this to represent a truly

incommensurate case, this is the 2D analog of the celebrated 1D “Aubry transition”

[29, 38, 31] here taking place at V0 crit ≃ 0.4 [39]. With this stipulation, for every

corrugation value V0 < V0 crit the colloid sliding over the incommensurate corrugation is

“superlubric”, and will take place for arbitrarily small applied force. By contrast, when


0 0.05 0.1 0.15

DC Force F

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

corr

ug

atio

n a

mp

litu

de

vcm

= 0

Shapiro steps unpinned sliding

static friction, immobile state

unpinned, dynamically sliding state

Aubry

tra

nsi

tion

V02 ∆

0

ρ = 14/15

Figure 5: DC Force F – corrugation amplitude V0 phase diagram for ρ = 14/15

mismatched colloids, in the absence of AC modulation (∆0 = 0). The red dots and line

represent the depinning line where at T = 0 the colloid abandons the statically pinned

state (shaded yellow region) in favor of a (nearly) free-sliding state for larger force F .

A truly incommensurate monolayer, where ρ is irrational, would exhibit a very similar

transition line, the 2D analogue of the 1D Aubry transition [31, 37]. Horizontal solid and

dot-dashed lines: a choice of corrugation (V0 = 0.5) and AC modulation ∆0 = 15%V0

amplitudes. In the force range for which both dot-dashed lines are in the shaded region,

the colloid layer is pinned. For large force, when both dot-dashed lines lie below the

depinning curve, the colloids advance freely. The intermediate force interval is the one

affected by the synchronization to the AC potential modulation, giving rise to Shapiro

steps.

V0 > V0 crit, the monolayer is pinned, and a finite static friction force F > 0 is required

to make the system slide [31]. The curve of Fig. 5 represents precisely this depinning

transition line for the ρ = 14/15 mismatched system, for nonzero and increasing force.

To understand what happens when we superpose an AC corrugation modulation

∆0 > 0 to the “phase diagram” of Fig. 5, we add horizontal dot-dashed lines representing

the maximum and minimum value covered by the overall corrugation magnitude during

an oscillation. As an example, in Fig. 5 we sketch V0 = 0.5 (horizontal solid line) and

∆0 = 15%V0 (horizontal dot-dashed lines). Synchronization between the washboard

sliding frequency and the AC modulation, and consequently Shapiro steps, can arise

naturally in the intermediate range of forces 0.02 ≤ F ≤ 0.15, where at each modulation

period the system crosses twice the pinned-unpinned transition curve. Figure 5 shows

that depending on V0 and ∆0, the F dependence can vary quite substantially. For

example, if V0 + ∆0 < V0 crit, then for any F the colloids will slide freely. For

V0 + ∆0 > V0 crit but V0 − ∆0 < V0 crit the initial pinned region (F ≤ 0.02 in figure)

would disappear: in this case the first Shapiro step could extend to arbitrarily small

applied force, by reducing the modulation frequency ν, as further discussed in Sec. 5.

This phase diagram can be taken as a guide for the choice of the simulation parameters.

In order to observe Shapiro steps, the colloid lattice must be pinned, e.g., V0 must be

above the critical corrugation V0 crit. The range of force where Shapiro steps exist widens


0 0.5 1

F (DC Force)

0

0.5

1

vcm

/(ν

ala

s)

∆0 = 60% V

0

1.35 1.4 1.45

0.95

1

1.05

1

0.1 0.2 0.30

0.02

0.04

0.06

0.08

w

w/2

w/3(b)

(a)

Figure 6: The Shapiro-step structure of the lattice mismatched ρ = 14/15 configuration

simulated at T = 0, with corrugation amplitude V0 = 1, modulation ∆0 = 60%V0, and

AC frequency ν = 0.02. (a) A zoom-in of the region of the “trivial” commensurate

step vcm/(ν alas) = 1 corresponding to all colloids advancing by alas at each period.

(b) A zoom-in of the region of the subharmonic steps, highlighting those matching

w = |1− ρ−1| = 1/14 and integer submultiples thereof.

out as the modulation amplitude ∆0 is increased. The phase diagram for ρ = 3/4 is

quite similar to that of ρ = 14/15, and in particular V0,crit ≃ 0.42 is nearly the same.

We now explore and describe the existence and nature of Shapiro steps in these

mismatched cases, adopting an arbitrary but reasonable corrugation V0 = 1 > V0,crit

and AC modulation ∆0 = 60%V0 (different parameters were explored too). Unlike the

ρ = 1 case, here a preliminary and careful relaxation of the initial force-free, equilibrium

structure is required. An initial simulation for a duration τ = 1000 is thus performed

with a time-independent corrugation (V0 turned on, ∆0 = 0), and no external force

applied (F = 0). Starting from the relaxed configuration, a sequence of force-driven

sliding simulations is run, now with the AC corrugation modulation turned on.

4.1. ρ = 14/15

4.1.1. Step structure. Let us consider ρ < 1, where it is useful to introduce as a misfit

measure the quantity

w =∣

∣

∣

∣

1− acollalas

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

1− 1

f

∣

∣

∣

∣

∣

, (10)

This quantity was found earlier to be the relevant parameter in the context of simulations

of mismatched crystalline systems in mutual contact, Refs. [40, 41, 42, 43, 44, 45, 46,

47, 48, 49, 50, 51]. The results of sliding simulation for w = 1/14, corresponding to

ρ = 14/15, are shown in Fig. 6. The Shapiro steps exist here too, but the mismatched

geometry exhibits a far richer pattern of Shapiro steps than the lattice-matched case:

we detect tens of steps in a much narrower force range than in the commensurate case

of Fig. 2.


First off, the regular integer plateau at vcm/(ν alas) = 1, at the same speed as the

first step in fully matched geometry, Fig. 3, is detailed in the inset Fig. 6a. Additionally,

Fig. 6 exhibits several other subharmonic steps at smaller driving force. It has long been

known that for mismatched lattices also fractional subharmonic Shapiro plateaus, i.e.

plateau velocities which are not integer multiples of ν alas [4, 52] are to be expected. In

particular the inset Fig. 6b zooms around a much lower speed, given byvcmν alas

= w . (11)

Here w = 1/14 and velocity steps appear at wm/n for several m and n values. All these

vcm < ν alas plateaus are subharmonic Shapiro steps. Unlike in the integer plateaus,

where all N particles advance by one (or several) lattice spacing(s) alas in one period

ν−1, at a fractional plateau characterized by vcm/(ν alas) = wm/n, a total fraction mwN

of all colloids advances by one lattice spacings during n modulation periods.

The quantization of sliding velocity to the value of Eq. (11) was discovered and

studied in the context of the sliding of mismatched crystalline systems in mutual contact,

and reported in the study of two sliding surfaces with an atomically thin solid lubricant

layer in between [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. There, the ratio ρ

was that of the lattice spacing of a “bottom” slider to that of the lubricant layer, w

was the mean velocity at which the lubricant moved over the bottom slider, in units

of the externally imposed speed of the “top” slider. The sliding of the lubricant with

relative speed w arose in that case because the vacancies (or interstitials) forming the

antisolitons (solitons) were the real entities dragged forward at the full speed of the top

layer, all (other) particles remaining essentially static. In the following we shall clarify

the relationship between the driven soliton dynamics in the sliding-friction models and

the subharmonic steps in the dynamics of colloids driven in a modulated-amplitude

potential.

4.1.2. Subharmonically sliding antisolitons. Figure 7 shows that the antisoliton lines

divide the cell into domains inside which the particles are essentially static at a minimum

of the corrugation potential. The overall forward sliding motion demanded by the

force F is actuated, for ρ < 1, through the backwards motion of the antisolitons,

where all vacancies are lumped. The antisolitons move to the left in synchrony

with the modulation, advancing faster when the instantaneous corrugation amplitude

V0+∆0 sin(ωt) of Eq. (7) is minimum. At that moment a fraction of colloids jumps from

the left to the right side of an antisoliton line, while all others remain essentially static.

In the wm/n step, this colloid displacement results in the full antisoliton pattern moving

by malas to the left every n modulation periods. Figure 8 illustrates the dynamics of a

single colloid. Most of the time the particle just oscillates, following the AC modulation:

when the amplitude V0+∆0 sin(ωt) is maximum, the particle is driven toward the nearest

corrugation-potential well; when the amplitude is minimum, the particle relaxes due to

the two-body repulsion. The amplitude of this oscillation is minimal when the colloid is

near the center of a domain, Fig. 8c, and increases as the colloid reaches the antisoliton,


Time

Figure 7: ρ = 14/15, T = 0, w/3 fractional plateau. The antisoliton pattern divides the

colloid lattice into domains or patches inside which colloids are approximately matched

to the static corrugation lattice. The underdense colloid monolayer slides rightward

due to the leftward motion of the antisoliton line, while particles inside the domains do

not slide. The specificity of this plateau is the behavior of particles at the antisoliton

boundary. Left and right are two successive snapshots separated by a single oscillation

period ν−1. At the end of this time interval, most colloids still occupy their initial

positions, but the 12 highlighted particles (red) crossed the antisoliton line to join the

next commensurate domain to the right. In the two following ν−1 periods the remaining

Nw− 12 = 16 particles will also cross, so that after the time nν−1 (here with n = 3) all

Nw = 28 particles will have crossed. Different plateaus with different w and different

subharmonic n will have an analogous but different pattern of traversing particles.

which is the domain boundary. A positive, uncompensated velocity spike, see Fig. 8a,b,

signals the moment when the particle crosses the gap and enters the neighboring domain,

Fig. 8d,e. It should be noted here that some fraction of the colloid rightward motion

also takes place while inside the commensurate domain, which is not rigid and thus

becomes progressively deformed elastically.

We are now in a position to clarify the nature of the w subharmonic steps. To

do this, we observe the colloidal pattern in a ’stroboscopic’ approach, by comparing

snapshots taken at successive times separated by one period, namely when the

corrugation amplitude, Eq. (7), acquires its maximum value V0 +∆0, i.e. for successive

half-odd integer values of νt. We highlight the particles that during the last period

ν−1 have underwent a displacement exceeding a certain threshold δ. Due to the elastic

domain deformations discussed above this threshold must be chosen appropriately: for

the conditions considered here a value 0.12 acoll ≤ δ ≤ 0.61 acoll provides consistent

results for all plateaus. The highlighted particles represent colloids, whose number we

designate as mi, which cross the antisoliton boundary between two neighboring domains

in that period i of oscillation. In the fundamental plateau where the quantized speed


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

xi(t)

200 400 600 800Time

0

0.02

0.04

vi(t)

(a)

(b)

(e)(d)(c)

Figure 8: ρ = 14/15, T = 0: The time evolution of the x component of (a)

velocity and (b) position of a single (highlighted in panels c-e) colloid, when F = 0.24

and the CM velocity is in the fractional Shapiro step of Fig. 6, characterized by

vcm/(ν alas) = w = 1/14. Because ρ < 1, the forward sliding motion is realized by a

backward motion of vacancies, here forming antisoliton lines. Velocity shows alternating

positive and negative spikes, plus a large positive peak when the leftward moving domain

boundary reaches, between t = 465 and t = 515, the colloid, which then jumps across

the antisoliton line, joining the next domain. Near the center of the domain – e.g. at

t ≃ 175, panel (c) – the lateral oscillation amplitude reaches its minimum, because the

particle sits close to a potential well bottom.

equals w, the pattern of advancing particles is the same at all oscillation periods, as

seen, e.g., in Fig. 9 and equals Nw where N is the total number of particles. In the

present case of Fig. 9 N = 392, w = 1/14, the Shapiro plateau has dimensionless velocity

w, and a total of Nw = 28 particles cross the boundary to the next domain at each

oscillation period. More specifically, 22 colloids at the edge of the island aligned along

y move straight in the x direction into the next corrugation minimum at a distance alas.

In addition, 12 more particles along the oblique antisoliton line advance with a sidewise

y component, thus proceeding into a minimum which is alas/2 to the right. The total

movement amounts therefore to effectively 22 + 12/2 = 28 = Nw particles advancing

by alas in each period. The very same picture applies to sliding a monolayer with any

different lattice-spacing ratio ρ = (1+w)−1. The fundamental subharmonic step occurs

again at velocity w.

The pattern of advancing colloids is somewhat more complex in the other

subharmonic steps. We explore these differences in Figs. 9 and 10. To exemplify, let

us consider again w = 1/14, but now focus on the w/2 plateau, Fig. 10. Here, a total


Figure 9: ρ = 14/15, N = 392 particles, V0 = 1, forward force F = 0.24, step

w = 1/14. Successive snapshot patterns of advancing colloids, separated by intervals of

one oscillation period ν−1. The antisolitons drift to the left, causing the CM to move to

the right. The particles whose displacements exceed a fixed threshold δ are highlighted

(lighter and bigger). In this step 22 particles shift to the right (i.e., along x) in each

oscillation period, crossing the vertical section of the antisoliton line, plus another 12

cross an oblique section, with a lateral y component of motion, and only half distance

along x. As a result a total of 22+12/2 = 28 particles advance by alas, joining the next

commensurate domain.

Figure 10: The subharmonic plateau w/2 for ρ = 14/15, w = 1/14, F = 0.185.

Successive patterns of advancing colloids, separated by intervals of one oscillation period

ν−1, illustrating the propagation of an antisoliton in this Shapiro plateau. In a first

displacement 10+4/2 = 12 (highlighted) particles jump (the 4 particles crossing oblique

antisolitons advance by a half lattice spacing along x). Then, at the next period, another

12+8/2 = 16 (highlighted) particles complete the advancement of the string of particles

bordering the antisoliton line. At successive periods, the alternating advancement of 12

and 16 particles repeats itself involving successive lines of colloids more and more to the

left.


particle number Nw again jumps between a domain and the next: but that takes place

in two successive oscillation periods instead of one as in the harmonic w step. In the

first AC oscillation period 10 + 4/2 = 12 colloids cross the antisoliton line, whereas the

remaining 12+8/2 = 16 jump in the second period. Similarly, in the step leading to the

w/3 plateau, it takes a sequence of three successive oscillation periods to complete the

advancement which transfers Nw particles across the antisoliton. The partial transfers

are in this case 10 + 4/2 = 12, 8 + 4/2 = 10, and 4 + 4/2 = 6 particles in period 1, 2

and 3 respectively.

In summary, each Shapiro step is characterized by a periodic advancement pattern

during which an integer multiple of Nw particles jump from a commensurate domain

to the next. However the m/n-subharmonic step of quantized velocity

vcmν alas

= wm

n(12)

is accomplished as a composite of n successive modulation periods, each lasting ν−1, of

the AC modulation. After one such complete migration, the entire pattern of positions

is displaced bodily by m lattice spacings alas. The subharmonic step leading to wm/n

is thus composite, involving n AC modulation periods. The total number mNw of

particles that cross the antisoliton is partitioned between the n periods,

mNw =n∑

i=1

mi . (13)

The detailed numbers mi of crossing particles at each individual AC period i will

generally depend on ρ, V0, ∆0, and on the angle between the sliding direction and

the antisoliton (soliton) orientation. Occasionally, the partition (13) may even change

along a subharmonic step as F increases. We have not found a general analytic rule, if

any, determining this partition.

4.1.3. Effect of the corrugation and modulation amplitudes. We now investigate the

role played by the corrugation amplitude V0 and by the modulation amplitude ∆0 in

the formation of Shapiro-step structures as exemplified in the ρ = 14/15 mismatched

configuration. First of all we remind ourselves that below the pinning threshold

corrugation amplitude V0 crit (which, as seen on Fig. 5 is around 0.4 for the present

case ρ = 14/15) the layer is superlubric, i.e., there is no static friction (other than that,

negligible, due to the weak 14/15 commensurability), and a time dependent modulation

with amplitude ∆0 such that V0+∆0 < V0 crit would produce no Shapiro steps. Thus we

concentrate on corrugation amplitudes above the threshold, where the particle lattice is

strongly pinned.

Figure 11 is useful to understand the corrugation dependence of Shapiro steps.

We probe three corrugation values: V0 = 0.5, V0 = 1 and V0 = 2 all with an

amplitude modulation ∆0 = 70%V0. The whole plateau structure moves to larger

forces when the corrugation V0 is increased. As expected from common sense, the

depinning force increases systematically with growing corrugation V0, nearly ∼ 20 times


0 0.2 0.4 0.6 0.8 1

F (DC Force)

0

0.05

0.1

0.15

0.2

0.25

vcm

/(ν

ala

s)

V0 = 0.5

V0 = 1

V0 = 2

w

w/2

w/3

Figure 11: ρ = 14/15: rescaled CM velocity for several corrugation amplitudes V0,

all with a modulation amplitude ∆0 = 70%V0 and frequency ν = 0.02. The initial

depinning force F increases rapidly with the corrugation amplitude V0. Shapiro steps

widen and shift to larger forces when the corrugation is increased.

larger when V0 changes from 0.5 to 2. With reference to Fig. 5, both V0 = 0.5 and

V0 = 1 have V0 − ∆0 < V0 crit, thus depinning occurs at an arbitrarily small force,

also depending on frequency as discussed below in Sect. 5. In contrast, V0 = 2 has

V0 − ∆0 = 0.6 > V0 crit, with a broad range of forces for which, independently of

frequency, the system remains pinned, thus explaining the huge increase of static friction.

The width of the Shapiro steps, especially the smaller ones, also increases with V0: for

example, the w/3 step, barely detectable for V0 = 0.5 and 1, becomes much broader

for V0 = 2. In general, a larger potential V0 amplitude, corresponding to a comparably

softer colloid lattice, spatially narrower solitons, and larger static friction, enhances the

Shapiro-step structure, which emerges more clearly.

Next, we explore the AC modulation amplitude dependence, Fig. 12, adopting

V0 = 1 and varying ∆0 from 40%V0, to ∆0 = 100%V0 (the purple triangles are indeed the

same data as the purple triangles of Fig. 11). As in the lattice-matched case Fig. 3, when

∆0 increases, the Shapiro steps become wider and extend further down to smaller force

F . Moreover, a larger AC modulation ∆0 depresses large-n longer-period subharmonic

w/n in favor of shorter ones with smaller n (e.g w/2 or w). Conversely, as Fig. 12

shows, all individual plateaus shrink and become more numerous for smaller modulation

amplitudes. Moreover, with ∆0 = 40%V0 we detect, in addition to w, w/2, w/3, and

w/5, extra fractional plateaus such as 4w/5, 2w/3. In short, a small AC modulation

amplitude ∆0/V0 produces many weak subharmonic steps, while a large modulation

promotes fewer stronger and wider ones.

4.2. ρ = 3/4

4.2.1. Step structure. There is a clear contrast between the ”nearly incommensurate”

case, such as ρ = 14/15, or w = 1/14, and a ”strongly commensurate” one, such as

ρ = 3/4, corresponding to w = 1/3, Eq. (10). In this strongly-commensurate case the


0 0.1 0.2 0.3 0.4

F (DC Force)

0

0.05

0.1

0.15

vcm

/(ν

ala

s)w

2w/3

w/2

w/3

∆0 = 100% V

0

∆0 = 70% V

0

∆0 = 40% V

0

Figure 12: Velocity of the colloid CM (rescaled as in Fig. 2), for ρ = 14/15, T = 0,

oscillation frequency ν = 0.02, corrugation amplitude V0 = 1, for different AC oscillation

amplitudes: ∆0 = 40%V0 (dots), ∆0 = 70%V0 (triangles), ∆0 = 100%V0 (squares).

Note that the smaller modulation ∆0 = 40%V0 exhibits a richer pattern of steps. The

width of the main subharmonic steps tends to increase with increasing modulation

amplitude ∆0, at the expense of minor steps.

0 0.1 0.2 0.3 0.4

F (DC Force)

0

0.2

0.4

0.6

0.8

1

vcm

/( ν

ala

s)

1

T = 0; ∆0 = 60% V

0

0.425 0.43 0.435

0.99

1.00

1.01

1

0.1 0.15

0.2

0.3

w

w/2(a)

(b)

Figure 13: ρ = 3/4, w = 1/3, N = 450 particles, corrugation V0 = 1, modulation

∆0 = 60%V0, oscillation frequency ν = 0.02. Two subharmonic Shapiro steps are

visible. Inset (a): Magnified view of the subharmonic plateaus. Inset (b): Magnification

of the “trivial” plateau, vcm/(ν alas) = 1 representing all colloids advancing as one every

modulation period.

Shapiro-step structure – Fig. 13 – shows fewer steps than in the nearly incommensurate

case of Fig. 6. Basically only two subharmonic plateaus (w and w/2) plus the “trivial”

step are detected. Moreover, the width of these plateaus is smaller than it was for

ρ = 14/15. However, as was the case there, even for strong commensurability ρ = 3/4

the plateau widths increase for increasing corrugation amplitude, and that even more

rapidly than for ρ = 14/15.


Figure 14: A stroboscopic picture of the ρ = 3/4 advancement pattern, similar to

Fig. 10, for the w/2 step. The antisoliton pattern is so weak that individual hexagonal

domains are nearly invisible. The advancing colloids (bolder particles) are highlighted:

150 particles advance in two sets at successive modulation periods: alternately 50 (left),

then 100 (center), then again 50 (right), and so on.

4.2.2. Subharmonic advancement of antisolitons. We examine the advancement

mechanism for the subharmonic plateaus in Figure 14. The mechanism is again similar

to that of Sec. 4.1, Fig. 10. Since ρ = 3/4 is further away from unity than 14/15, the

antisoliton pattern is composed by much closer, partly overlapping and less distinct lines.

The subdivision in domains is nearly invisible, with each particle always approximately

as close to the edge of a domain as to its center. As a result, the oscillations around

the average colloid trajectory are far less evident than in Fig. 8. Nevertheless, the

advancement mechanics is the same as in the earlier case: e.g. the w/n subharmonic

step involves Nw = 150 particles, advancing during n = 2 periods: we identify a clear

pattern of advancing particles, displayed in Fig. 14. Here 150 colloids at the domain

boundary advance in a first group of 50 particles in one modulation period, then a

second group of 100 in the next period. In the w step, all 150 particles at the boundary

cross the antisoliton line at every period: they could be identified by superimposing the

highlighted colloids in the first and second panels of Fig. 14.

4.2.3. Thermal effects. Finally we explore thermal effects on the subharmonic step

structure. All tests in the ρ = 14/15 system show that at finite temperature the

fragile Shapiro structure is strongly affected: the steps are washed away by the colloidal

diffusive motion. No clear step can be detected for T > 0.001 ≪ Troom = 0.04, whether

the standard V0 = 1 and ∆0 = 60%V0 or other parameters are adopted.

By contrast, with the ρ = 3/4 mismatch ratio, where commensurability is stronger,

the effects of thermal fluctuations are less dramatic, and at least the strongest steps

survive. As shown in Fig. 15, the w plateau persists up to and above Troom, the thermal

effects leading to a slight reduction in width and to a slight edge rounding compared

to T = 0. We explored even higher temperature, and found that even at T = 4 Troom

the w step leaves some remnant of its existence in the force-velocity diagram, Fig. 15a.

However, even room-temperature fluctuations are enough to cancel out completely the


0 0.05 0.1 0.15

F (DC Force)

0

0.1

0.2

0.3

0.4

vcm

/(ν

ala

s)

Troom

T = 0

0.13 0.14 0.15

0.32

0.34

0.36

(a)

Troom

4 Troom

w

w/2

Figure 15: Thermal effects in the ρ = 3/4 mismatched configuration, for V0 = 1,

∆0 = 60%V0. Like in Fig. 4, the mean value and error bar of vcm are computed by

averaging over several simulations. The w step remains quite well identified at Troom,

while the w/2 step disappears altogether. (a): a zoom-in of the subharmonic w step at

T = Troom compared with the much larger temperature T = 4 Troom where the plateau

structure is washed out by thermal noise.

100 1000Particle Number N

0.0001

0.001

0.01

Sta

nd

art

Dev

iati

on

F = 0.08, off step

F = 0.145, in step

power law N -1/2

Figure 16: Size scaling of the fluctuations of the CM velocity in room-temperature

simulations of the ρ = 14/15 mismatched configuration for V0 = 1, ∆0 = 60%V0,

executed for F = 0.08 (climb between steps) and F = 0.145 (step w) of Fig. 15. The

various sizes are simulated with rectangular cells similar to the one depicted in Fig. 1.

The fluctuations are obtained as the standard deviation evaluated over 4 independent

simulations for each size. Dashed line: a N−1/2 guide to the eye.

w/2 plateau.

Thermal effects on Shapiro steps also depends to some extent on size. As long

as the colloid density remains uniform, thermal fluctuations average out for increasing

sample size and averaging time: for a given temperature, a larger sample size and longer

simulation show flatter, sharper plateaus. Figure 16 shows how the thermal fluctuations

on the colloid CM speed depend on the number N of colloids in the simulation supercell.

For large size, as expected by statistics, thermal fluctuations scale as N−1/2, while when


0.001 0.01Frequency ν

0.01

0.1

1

Dep

inn

ing

Fo

rce

Fdep

in

ρ=1

ν2/3

ρ=14/15, w step

ρ=14/15, w/2 step

ρ=14/15, w/3 step

ν1

Figure 17: The static depinning force as a function of the modulation frequency in

conditions where the sliding barrier vanishes during the modulation period. Circles: the

lattice-matched geometry, with V0 = 1, ∆0 = 100%V0. Other symbols: ρ = 14/15, with

V0 = 1, ∆0 = 70%V0.

the cell lateral size is comparable to or smaller than the correlation length associated

to the colloid-colloid interaction, dynamical correlations enhance the fluctuations. Note

that inside a plateau (triangles in Fig. 16), the fluctuations are much smaller and the

correlation length (marked by the knee in the size scaling) significantly longer, than in

a climb between steps.

5. The depinning force

This section deals with the DC-Force – corrugation-amplitude diagram, Fig. 5. We

can foresee that, depending on V0 and ∆0, the static-friction force Fdepin, namely the

minimum force required to maintain a net nonzero sliding speed, can fall in one of two

alternative regimes. The simplest condition occurs when V0 − ∆0 exceeds the Aubry

V0 crit: then the colloid remains statically pinned throughout the modulation period, with

the result that sliding requires a finite driving force greater or equal than the depinning

force appropriate to a corrugation V0 −∆0 in the absence of modulation.

Conversely, when V0 −∆0 < V0 crit, for a fraction ffree of the oscillation period ν−1,

the pinning barrier vanishes, and the colloid slides freely. A slow enough AC modulation

gives the particles enough time to take advantage of the temporary lack of pinning and

generate the first Shapiro step for arbitrarily small driving force F . In such condition,

the depinning force is a function of the modulation frequency. The depinning force Fdepin

can be estimated as follows: During the unpinned time ffreeν−1 the colloids advance at

a speed ≃ µF , where µ ≃ η−1 is the colloid mobility; by equating the displacement

ffreeν−1Fµ to a minimum advancement compatible with the overall displacement in the

lowest Shapiro step, say wnalas, we obtain

Fdepin ≃ walasnµffree

ν . (14)


Thus, under modulation conditions such that V0 −∆0 < V0 crit, the static friction force

is predicted to vanish linearly with the modulation frequency. Simulations confirm this

power law scaling, indicated by the dashed line in Fig. 17. For decreasing frequency,

simulations also show that the n > 1 subharmonic steps tend to weaken and eventually

disappear, see e.g. the diamonds (w/2) and triangles (w/3) in Fig. 17. As a result,

eventually at very low frequency depinning occurs directly into the w step.

The fully-matched case ρ = 1 discussed in Sec. 3 has no Aubry transition, so

V0 crit = 0. Even so, at 100% modulation rate (∆0 = V0), in the brief instants when

sin(ωt) ≈ −1, again free sliding is allowed. At finite driving force F a finite time interval

ffreeν−1 is compatible with barrier overcoming and consequent free sliding. This interval

is obtained by evaluating the region of F ≥ Fs1 using the expression (9) (under the

condition ∆0 = V0) for the barrier crossing force, obtaining

ffree =1

2+

1

πarcsin

(

9Falas8πV0

− 1)

. (15)

By expanding the arcsin function with its argument near −1 (small driving force) we

obtain ffree ≃ [9Falas/(4π3V0)]

1/2. As above, we then equate the displacement ffreeν−1Fµ

to the minimum advancement required for the colloids to advance by one corrugation

lattice spacing, say 12alas, obtaining

Fdepin ≃ π

(

V0alas9µ2

)1/3

ν2/3 . (16)

Again, simulations confirm this result, see Fig. 17, with depinning always occurring into

the trivial harmonic step. In principle a similar “critical” ν2/3 power-law regime is to

be expected even in the mismatched case ρ < 1, under the condition V0 − ∆0 = V0 crit

of an instantaneous touching of the free-sliding superlubric Aubry phase during each

modulation cycle.

6. Discussion and Conclusion

The present work addresses, by means of MD simulations, the question whether Shapiro

step structures can occur in the forced sliding of a 2D colloidal monolayer immersed in an

optical lattice spatially periodic corrugation potential whose amplitude has a static DC

component plus an AC modulation that oscillates sinusoidally in time. We confirm that

Shapiro steps should appear, in the form of intervals of static force within which the CM

colloid velocity is quantized, at least at zero temperature, to a value entirely determined

by the modulation frequency and by geometric parameters. The only condition required

for this to occur at T = 0 is that in the absence of the AC modulation the colloids should

be pinned by static friction, and not free sliding, as can happen when the 2D colloid

lattice and the optical lattice are mismatched and the optical lattice is sufficiently strong.

In commensurate, perfectly lattice-matched conditions ρ = 1 only ”trivial” integer steps

are expected, corresponding to the synchronized periodic advancement of all particles at

each AC modulation period. In lattice-mismatched conditions ρ 6= 1 we find additional


subharmonic Shapiro plateaus beside the integer ones. The crucial quantity governing

the subharmonic plateaus is w = |1−ρ−1|, which determines the quantized velocity step

value in the totally general form

vcm = ν alaswm

n, (17)

where ν is the AC frequency, alas is the optical lattice spacing, and m and n are small

integers. Interestingly, we find that every quantized Shapiro step has a specific real-

space signature corresponding to which particles advance synchronously with the AC

modulation – a signature which is purely geometric. In a step the total number of

advancing particles per modulation period equals exactly a fraction N wm/n of the

total particle number N . However, a step corresponding to integer n lasts n periods.

During each period ν−1 only a sub-fraction of these particles advance by one lattice

step, and the total Nwm is achieved only at the end of a cycle of n periods.

We analyzed how the corrugation amplitude V0 and modulation ∆0 influence the

ensuing step structure. An increase in V0 leads to a comparably softer colloidal layer,

thus increased localization and static friction, resulting in a shift to higher forces F of

the Shapiro plateaus and generally an increase in their width. On the other hand an

increase in ∆0 to the point of making V0 − ∆0 < V0 crit leads the oscillation to explore

more and more of the hard-colloidal-layer region for part of the oscillation period. In

this region, free sliding is possible, as well know in the 1D FK model [38]. This results

in more extended steps, in particular down to smaller F . In such cases, the smallest

(depinning) force is a function of the modulation frequency, and we verified that it

changes either as ν as ν2/3 depending on whether a free sliding state is crossed for a

finite fraction of the modulation period or just touched instantly. Additionally, a very

large ∆0 ≃ V0 tends to simplify the rich ladder of subharmonic step structures observed

at intermediate ∆0.

We also checked how important the specific shape of the corrugation-amplitude time

modulation really is for the possibility to observe Shapiro steps. We ran simulations

both for the lattice-matched case and for the ρ = 3/4 mismatch ratio, replacing sin(ωt)

in Eq. (7) with a square wave sign(sin(ωt)), where sign(x) equals x/|x| for nonzero

x. Compared to the sinusoidal case with identical modulation amplitude ∆0 and

other conditions, the square-wave simulations show quite similar velocity-force profiles,

with marginally wider and more robust Shapiro steps. This small difference can be

understood as a consequence of the larger RMS modulation of the square wave than the

sinusoidal wave for the same peak-to-peak amplitude.

Thermal fluctuations and density inhomogeneities present in real colloid sliding

experiments are likely to affect the possibility to observe Shapiro structures. While

the perfectly-matched system is barely affected by the thermal Brownian motion, the

subharmonic Shapiro-step structures of mismatched systems are far more fragile. Near

matching between the colloid and the optical lattice, the loose soliton network is strongly

affected by thermal fluctuations, which wash easily away most of the delicate Shapiro

plateaus. By contrast, for a more robust (but commensurate) lattice mismatch such as


ρ = 3/4, we retrieve subharmonic steps up to Troom and even above. We test also the

size dependence in the latter case: as long as the density is uniform, a greater number

of particles averages out thermal fluctuation, ending up in a flatter and less rounded

step.

The model explored in the present work neglects hydrodynamic interactions [53]:

while such velocity-dependent many-body forces may affect the quantitative detail of

the Shapiro steps, we do not expect they would change the qualitative picture much. In

particular, nonuniform solitonic motion, as occurs in mismatched configurations, could

be enhanced by hydrodynamic interactions. We are currently investigating this issue in

quantitative detail. Also, the detail of the two-body interaction, Eq. (5), is not likely

to play any qualitative role in the Shapiro-steps physics. In particular, we expect a

similar behavior if the colloid-colloid interaction, rather than screened electrostatic, was

the power law describing the repulsion between magnetic colloidal particles in a layer,

controlled by a perpendicular magnetic field [54, 55, 56].

In conclusion, this theoretical study predicts that Shapiro-like plateau structures of

colloid CM velocities should be easily detected experimentally in a 2D rich geometry– at

least in lattice-matched condition, with potential amplitude and oscillation modulation

values varying over broad ranges. In mismatched conditions, on the other hand, the

experimental detection of subharmonic plateaus seems limited by the ability to provide

large regions where the 2D colloid density is constant in space with sufficient accuracy.

While experimentally this may not be possible for all subharmonic quantizations, still

for a lattice-spacing ratio sufficiently distant from unity, a step structure should be

detectable under experimental conditions. Subharmonic steps were indeed detected

(although not identified as such) in the simpler 1D experimental setup of Ref. [11].

Acknowledgments

We acknowledge useful discussion with Clemens Bechinger. This work was partly

supported under the ERC Advanced Grant No. 320796-MODPHYSFRICT, by the

Swiss National Science Foundation through a SINERGIA contract CRSII2 136287, by

PRIN/COFIN Contract 2010LLKJBX 004, and by COST Action MP1303.

References

[1] Ya. I Frenkel and T. A. Kontorova, Phys. Z. Sowietunion 13, 1 (1938).

[2] T. A. Kontorova and Ya. I. Frenkel, Zh. Eksp. Teor. Fiz. 8, 89 (1938).

[3] T. A. Kontorova and Ya. I. Frenkel, Zh. Eksp. Teor. Fiz. 8, 1340 (1938).

[4] L. M. Florıa and J. J. Mazo, Adv. Phys. 45, 505 (1996).

[5] K. Mangold, P. Leiderer, and C. Bechinger, Phys. Rev. Lett. 90, 158302 (2003).

[6] S. Bleil, H. H. von Grunberg, J. Dobnikar, R. Castaneda-Priego, and C. Bechinger, Europhys.

Lett. 73, 450 (2006).

[7] T. Bohlein, J. Mikhael, and C. Bechinger, Nat. Mater. 11, 126 (2012).

[8] T. Bohlein and C. Bechinger, Phys. Rev. Lett. 109, 058301 (2012).

[9] A. Vanossi and E. Tosatti, Nature Mater. 11, 97 (2012).


[10] A. Vanossi, N. Manini, and E. Tosatti, P. Natl. Acad. Sci. USA 109, 16429 (2012).

[11] M. P. N. Juniper, A. V. Straube, R. Besseling, D. G. A. L. Aarts, and R. P. A. Dullens, Nature

Commun. 6, 7187 (2015).

[12] S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).

[13] B. D. Josephson, Phys. Lett. 1, 251 (1962).

[14] W. C. Stewart, App. Phys. Lett. 12, 277 (1968).

[15] D. E. McCumber, J. App. Phys. 39, 3113 (1968).

[16] R. L. Kautz, Rep. Prog. Phys. 59, 935 (1996).

[17] M. Kvale and S. E. Hebboul, Phys. Rev. B 43, 3720 (1991).

[18] A. B. Kolton, D. Domınguez, and N. Gronbech-Jensen, Phys. Rev. Lett. 86, 4112 (2001).

[19] N. Kokubo, R. Besseling, and P. H. Kes, Phys. Rev. B 69, 064504 (2004).

[20] M. Eichberger, H. Schafer, M. Krumova, M. Beyer, J. Demsar, H. Berger, G. Moriena, G. Sciaini,

and R. J. D. Miller, Nature 468, 799 (2010).

[21] J. Tekic and B. Hu, Appl. Phys. Lett. 95, 073502 (2009).

[22] J. Tekic and B. Hu, Phys. Rev. E 81, 036604 (2010).

[23] J. Tekic and Z. Ivic, Phys. Rev. E 83, 056604 (2011).

[24] W. Mu, Z. Liu, L. Luan, G. Wang, G. C. Spalding, and J. B. Ketterson, New J. Phys. 11, 103017

(2009).

[25] P. Mali, J. Tekic, Z. Ivic, and M. Pantic, Phys. Rev. E 86, 046209 (2012).

[26] A. Libal, C. Reichhardt, Janko, and C. J. Olson Reichhardt, Phys. Rev. Lett. 96, 188301 (2006).

[27] M. P. Allen and D. J. Tildesley, Computer Simulations of Liquids (Oxford University Press, Oxford,

1991).

[28] D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2457 (1979).

[29] O. M. Braun and Yu. S. Kivshar, The Frenkel-Kontorova Model: Concepts, Methods, and

Applications (Springer, Berlin, 2004).

[30] A. Vanossi and O. M. Braun, J. Phys.: Condens. Matter 19, 305017 (2007).

[31] D. Mandelli, A. Vanossi, M. Invernizzi, S. V. Paronuzzi Ticco, N. Manini, and E. Tosatti,

arXiv:1508.00147, submitted to Phys. Rev. B (2015).

[32] A. D. Novaco and J. P. McTague, Phys. Rev. Lett. 38, 1286 (1977).

[33] J. P. McTague and A. D. Novaco, Phys. Rev. B 19, 5299 (1979).

[34] H. Shiba, J. Phys. Soc. Jpn. 46, 1852 (1979).

[35] H. Shiba, J. Phys. Soc. Jpn. 48, 211 (1980).

[36] D. Mandelli, A. Vanossi, N. Manini, and E. Tosatti, Phys. Rev. Lett. 114, 108302 (2015).

[37] S. Aubry and P. Y. Le Daeron, Physica D 8, 381 (1983).

[38] M. Peyrard and S. Aubry, J. Phys. C: Solid State Phys. 16, 1593 (1983).

[39] The Aubry transition occurs rigorously only for strictly incommensurate (irrational length ratio

f) conditions. In rationally matched situations such as those considered here, the “superlubric”

state is characterized by an extremely small nonzero static friction, which vanishes exponentially

with the distance from the Aubry point V0 crit.

[40] A. Vanossi, N. Manini, G. Divitini, G. E. Santoro, and E. Tosatti, Phys. Rev. Lett. 97, 056101

(2006).

[41] N. Manini, A. Vanossi, G. E. Santoro, and E. Tosatti, Phys. Rev. E 76, 046603 (2007).

[42] N. Manini, M. Cesaratto, G. E. Santoro, E. Tosatti, and A. Vanossi, J. Phys.: Condens. Matter

19, 305016 (2007).

[43] G. E. Santoro, A. Vanossi, N. Manini, G. Divitini, and E. Tosatti, Surf. Sci. 600, 2726 (2006).

[44] M. Cesaratto, N. Manini, A. Vanossi, E. Tosatti, and G. E. Santoro, Surf. Sci. 601, 3682 (2007).

[45] A. Vanossi, G. E. Santoro, N. Manini, M. Cesaratto, and E. Tosatti, Surf. Sci. 601, 3670 (2007).

[46] A. Vanossi, N. Manini, F. Caruso, G. E. Santoro, and E. Tosatti, Phys. Rev. Lett. 99, 206101

(2007).

[47] A. Vanossi, G. E. Santoro, N. Manini, E. Tosatti, and O. M. Braun, Tribol. Int. 41, 920 (2008).

[48] N. Manini, G. E. Santoro, E. Tosatti, and A. Vanossi, J. Phys.: Condens. Matter 20, 224020


(2008).

[49] I. E. Castelli, N. Manini, R. Capozza, A. Vanossi, G. E. Santoro, and E. Tosatti, J. Phys.: Condens.

Matter 20, 354005 (2008).

[50] I. E. Castelli, R. Capozza, A. Vanossi, G. E. Santoro, N. Manini, and E. Tosatti, J. Chem. Phys.

131, 174711 (2009).

[51] A. Vigentini, B. Van Hattem, E. Diato, P. Ponzellini, T. Meledina, A. Vanossi, G. Santoro, E.

Tosatti, and N. Manini, Phys. Rev. B 89, 094301 (2014).

[52] F. Falo, L. M. Florıa, P. J. Martinez, and J. J. Mazo, Phys. Rev. B 48, 7434 (1993).

[53] H. Diamant, B. Cui, B. Lin, and S. A. Rice, J. Phys: Cond. Matter 17, S2787 (2005).

[54] E. L. Bizdoaca, M. Spasova, M. Farle, M. Hilgendorff, and F. Caruso, J. Magn. Magn. Mater. 240,

44 (2002).

[55] X. Xu, G. Friedman, K. D. Humfeld, S. A. Majetich, and S. A. Asher, Chem. Mater. 14, 1249

(2002).

[56] J. Ge and Y. Yin, Adv. Mater. 20, 3485 (2008).

monolayersinopticallattices arxiv:1508.06201v2 [cond-mat.mes … · 2016. 9. 1. · e-mail:...

Documents